moj seminar P ... - F9

8 Representative calculation: nonradiative energy transfer as a result of ion-ion electric dipole interaction4 Representative calculation: nonradiative energy transfer as a result of ionionelectric dipole interactionSince the gain of a laser depends directly on the concentration of lasing centres, it would appear to be beneficialto increase this concentration to as high level as possible. Yet, optimal doping levels of different types of lasercrystals vary between a few hundreds of a percent and several tens of percents. One limitation is posed bythermal lensing - see section 5 - , because the thermal loading in a crystal increases with doping concentration.The other limitation arises from the ion-ion interactions. The interaction strength is a function of the separationof the two ions and the physical mechanisms of interaction [2]. Usually the doping level is not so high toallow the ions to couple strongly, rather the ions maintain their independent optical properties and interact bythe transfer of energy from one to another through nonradiative processes such as multipolar interactions; orthey may be coupled indirectly through the radiative emission from one ion to the other. In this paper only thefirst case, the weak ion-ion radiationless coupling, more precisely dipole-dipole resonant interaction leading toenergy transfer between ions with individual energy levels involved remaining the same as those for isolatedions, will be dealt with.Whatever the type of ion-ion interaction mechanisms, it can have very different effect on the propertiesof the laser performance. It can either enhance or degrade a specific laser performance characteristic. In thissection the general form of weak ion-ion radiationless coupling will be presented, and in section 6.1 one of theeffects - sensitized pumping - will be described.We employ the sensitizer-activator scheme from page 5. The photon energy absorbed by sensitizer moves[7], through the dipole-dipole interaction possibly aided by surrounding lattice relaxation, to the activator inone step without radiation exchange between sensitizer and activator ion. As with any quantum-mechanicalprocess, the rate for energy transfer to occur is described by the golden rule, given byW sa = 2π |M sa| 2 ρ f (E f = E i ) ; (4.1a)M sa = 〈 〈 ∣∣ψ ∣∣H ion-ion〉 ∑ ψf ∣∣H ion-ion〉 〈 ∣∣intψ j ψj ∣∣H ion-ion〉∣intψ if ∣intψ i ++ . . . (4.1b)Eji − E j}{{} }{{}resonant interactionphonon-assisted energy transferIn the second line, the matrix element M sa for the transition was expanded according to time-dependentperturbation theory: the first term describes a resonant (direct) interaction while the higher-order terms describephonon-assisted energy transfer. The former is discussed below, while the theory including the latter term can befound in literature [7, 2]. Hintion-ion is the ion-ion interaction Hamiltonian and ψ i,f,j represents the antisymmetrizedproduct functions of the optically active electrons on the sensitizer and activator ions.Under these circumstances, the main interaction is electromagnetic Coulomb interaction between the electronsof the sensitizer and of the activator ion. This interaction can be expressed as a multipole expansion usinga Taylor’s series about the sensitizer-activator separation ⇀ R sa:H EMint= −e 2∣4π εε ∣∣∣⇀0 r ∣∣ =sae 2= − ∣ ∣∣∣∣⇀ ∣∣∣ =4π εε 0 R sa + ⇀ r a − ⇀ r s= −e 24π εε 0 R 3 sa(⇀r s · ⇀r a − 3R 2 sa(⇀r s · ⇀R sa) (⇀r a · ⇀R sa))+ . . . (4.2)Only the first term in expansion, representing electric dipole-dipole interaction is significant; higher termsinclude quadrupole and higher order interactions. The dielectric constant of the host crystal is given by ε. Thevectors ⇀ r a and ⇀ r s designate the positions of the electrons on the sensitizer and activator as shown in Fig. 7.